第 4 章 数值计算 (1)
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第 4 章 数值计算 (1). 主要内容：. ① 数据分析； ②矩阵分析； ③多项式运算； ④数值插值和拟合； ⑤数值积分和微分 。. 4.1 概述. 数值计算的特点： 1 ）数值计算是定义在数值数组的基础上，在计算之前必须定义变量并赋值。 2 ）数值计算有误差。 3 ）数值计算的计算速度较快。 4 ）数值计算除定义了算术运算、关系运算和逻辑运算三种，还提供了许多运算函数。. 4.2 数据分析. 4.2.1 统计分析 1 最大值和最小值 1 ）求向量的最大和最小值. - PowerPoint PPT PresentationTRANSCRIPT
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4.1 1234
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4.2 4.2.11 1y=max(X) XyX[y,I]=max(X) XyIXXmin(X)max(X)
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4-1 x=[-43,72,9,16,23,47] xy=max(x) %x[y,l]=max(x) %xz=min(x) %x[z,m]=min(x) %x
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2Amax(A) iAi [Y,U]=max(A) YUYAUmax(A,[],dim) dim12dim1max(A)dim2iAiminmax
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4-2
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3U=max(A,B) A,BUA,BUA,BU=max(A,n) nUAUAnminmax
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4-3 x=[4 5 6;1 4 8]; y=[1 7 5;4 5 7]; p=max(x,y) ; P23xyp
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2 meanmedianmean(X) Xmedian(X) Xmean(A) iAi
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median(A) iAimean(A,dim) dim1mean(A)dim2iAimedian(A,dim) dim1median(A)dim2iAi
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4-4 x=[1 8 4 2;9 6 2 5;3 6 7 1] median(x) median(x,1) %median(x,2) %mean(x) mean(x,1) %mean(x,2) %
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3 sum(X) Xprod(X) X Asum(A) iAiprod(A) iAi
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sum(A,dim) dim1sum(A)dim2iAiprod(A,dim) dim1prod(A)dim2iAi
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4-5 x=[4 5 6;1 4 8]x sum(x)sum(x,1) %sum(x,2) %sum(sum(x)) %
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prod(x)prod(x,1) %prod(x,2) %prod(prod(x)) %
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4 cumsum(X) Xcumprod(X) Xcumsum(A) iAi
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cumprod(A) iAicumsum(A,dim) dim1cumsum(A)dim2iAicumprod(A,dim) dim1cumprod(A)dim2iAi
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4-6 a = [1 2 3; 3 9 6; 4 10 8; 4 0 7]a cumsum(a) %cumsum(a,2) %cumprod(a)cumprod(a,1) %cumprod(a,2) %
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xstd(xflag)flag01Astd(A)A Y=std(A,flag,dim) dim12dim=1dim=2flag01flag=0dim=15
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4-7 a = [1 2 3; 3 9 6; 4 10 8;4 0 7]std(a)std(a,0,1) %astd(a,1,1) %astd(a,0,2) %astd(a,1,2) %a
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6 sort(X) X [Y,I]=sort(A,dimMODE) A Y I YA dim A MODE
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sortrows B = sortrows(A) A A B = sortrows(A,column) [B,index] = sortrows(A) A B = A(index) A m-by-n B = A(index,:)
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4-8 a = [1 2 3; 3 9 6; 4 10 8; 4 0 7] sort(a,1)sort(a,1,'ascend') %asort(a,2)sort(a,2,'ascend') %asort(a,1,'descend') %asort(a,2,'descend') %a
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4-9 a=['one ','two ','three ','four ','five']; sortrows(a)
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4.2.2 cov(x) x cov(A) A corrcoef(X) X
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4-10 xy xyx=[1 2 3 4 5]; y=[2 4 6 8 10];cx=cov(x)cy=cov(y) cxy=cov(x,y)
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4.2.3 fft fft2 fftn n fftshift Ifft ifft2 ifftn n ifftshift fftshift
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4-11 00.1 0.2 3.14
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4.3 4.3.1 1 A AA
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4-12 a=[1 2 3];b=a %a=[1; 2; 3];b=a %a=[1 2; 2 3; 3 4];b=a %a=[1+2*i 3-4*i];b=a %
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2 X = diag(v,k) v nkn+abs(k)k = 0k > 0 k < 0v = diag(X,k) k k = 0
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diag diag(diag(A)) diagAdiag4-13 n
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3 flipudfliplr B = flipud(A) B = fliplr(A)
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flipdim B = flipdim(A,dim) dim =1dim =2flipdim(A,1) flipud(A), flipdim(A,2) fliplr(A)
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4-14 a=[1 2 3 4;5 6 7 8;9 10 11 12;13 14 15 16]; flipud(a)%flipdim(A,1) %fliplr(a)%flipdim(A,2) % a=[1; 2; 6; 7; 8]; flipud(a) %a=[1 2 6 7 8];flipud(a) %
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4 B = rot90(A) A90B = rot90(A,k) Ak90k4-15 a=[1 2; 6 7; 8 0] rot90(a) rot90(a,2) rot90(a,3)
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5 B = reshape(A,m,n) Amn BAmn 4-16 12 3426
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6 tril (X) X tril (X,K) XKK =0K >0K< 0
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triu (X) X.triu (X,K) XKK =0K >0K< 0
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4-17 3
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4-18 a=-4:4 A=reshape(a,3,3) %a a1=diag(A,1) %A A1=diag(a1,-1)%a1 flipudA=flipud(A) %AfliplrA=fliplr(A) %Arot90A=rot90(A) %90A
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4-19 22x,yz 143xyz2x,yz
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4.3.2 1 inv(A)A:pinv(A)4-20 a=[-1 2 0;-2 3 0;3 0 2]; inv(a)
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4-21
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A=[2 2 -1 1;4 3 -1 2;8 3 -3 4;3 3 -2 -2];b=[4 6 12 6]'; x=inv(A)*b x = 0.6429 0.5000 -1.5000 0.2143 0.6429 0.5000 -1.5000 0.2143
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4-22 XAXB = CA=[1 2 3;2 2 1;3 4 3];B=[2,1;5 3];C=[1 3;2 0;3 1];X=A\C/B
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4-23
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2 rank(A) A4-24 1a=[4 2 -6;7 5 4 ;3 4 9];R=rank(a)2a=[5 4 -2 ;4 5 2;-2 2 8] ; R=rank(a)
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3 det(A)A4-25 n=input('please input the order of magic matrix');a=magic(n);det(a)
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4-26 4-24 qiunifunction ii=qiuni(a)%a matrixad=det(a)if ad==0 disp('no inverse') ii=pinv(a);else ii=inv(a);enda=[4 2 -6;7 5 4 ;3 4 9]; ii=qiuni(a)
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4 trace(A) Asum(diag(A))4-27 1trace(magic(3))ans = 152sum(diag(trace(magic(3))))
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5 1) norm (x) x2 norm (x,1) x1 norm (xinf) x max(abs(A)) norm (x-inf)xmin(abs(A))2) norm (A,1) 1 norm (A,2) 2 norm (A,inf)
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4-28 ( 0 1 2 3)x = [0 1 2 3] ;norm(x,1) %x1norm(x) %x2norm(x,inf) %xnorm(x,-inf) %x-
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4-29 A=[2 2 -1 1;4 3 -1 2;8 3 -3 4;3 3 -2 -2] ;norm(A,1) %A1norm(A) %A2norm(A,inf) %A
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6 cond(A1) A1-cond(A)cond(A2) A2-cond(Ainf) A-
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4-30 A=[2 2 -1 1;4 3 -1 2;8 3 -3 4;3 3 -2 -2] ;cond(A,1) %A1cond(A) %A2cond(A,inf)%A
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7 R = rref(A)A8 z = null z z = null(A,r)zAX=0
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9 subspace (A,B) A,B4-36 H=magic(8);A=H(:,2:4);B=H(:,5:8);theta=subspace(A,B)
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4.3.3 eig(A) A [X,D]=eig(A) ADXAX=XD
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4-37
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4.3.41 LU [L,U]=lu(A) ULLU=A [L,U,P]=lu(A) ULPLU=PA
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4-38 LU[L,U]=lu(magic(3))L*U
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4-39
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2 Cholesky chol(A) ACholeskyA [G,err]=chol(A) ACholeskyGAerr
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4-40 Cholesky a=[2 5 4;5 2 1;4 1 8]; chol(a)
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3 qr [Q,R]=qr(A) mmQRQQRA=QR[Q,R,P]=qr(A) QRPQRAP=QR
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4-41 qr
a=[2 5 4;5 2 1;4 1 8];[Q,R]=qr(a)[Q,R,P]=qr(A)
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4 SVDsvd svd(A) A [U,S,V]=svd(A) SAmmnnUVA=USVUAV=SS
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4-41 a LUQR Chollesky a=[9 1 2;5 6 3;8 2 7];[v,d]=eig(a)a=[9 1 2;5 6 3;8 2 7];[u,s,v]=svd(a) [l,u]=lu(a)[q,r]=qr(a)c=chol(a)
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